Elsner was born in Nuremberg, Middle Franconia. In 1959, she went to Vienna to study philosophy, Germanic letters and drama.
Elsner then lived as a freelance writer in various places: Lake Starnberg, Frankfurt, in Rome from 1963 to 1964, in London from 1964 to 1970, then in Paris, Hamburg, New York, and finally in Munich.
She was among the members of Gruppe 47, which also included Günter Grass and Heinrich Böll.
In her 1970 novel Berührungsverbot, several couples try to transcend the limits of the bourgeois sexual mores of their middle-class background by engaging in group sex orgies. In Switzerland, a journal that published excerpts from the novel was banned, and in Austria it was attacked as harmful to children.
Elsner described herself as a Leninist. She was a long lasting member of the German Communist Party.
Elsner committed suicide by jumping out of a window, in Munich, on 13 May 1992.
Here we have Elsner at her best - the usual hilarious, caustic wit but without the alienating stylistic idiosyncracies of her early, "experimental" years. In this contemporary reworking of Madame Bovary, we witness what could well be Elsner's only instance of sympathizing with a character. I loved Abseits; I used to read it once a year, suggested it as reading for several Germanistik courses, gave away countless copies and could quote from it extensively.